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Paul

January 27, 2020

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### Section 7-7 : Types of Infinity

Most students have run across infinity at some point in time prior to a calculus class. However, when they have dealt with it, it was just a symbol used to represent a really, really large positive or really, really large negative number and that was the extent of it. Once they get into a calculus class students are asked to do some basic algebra with infinity and this is where they get into trouble. Infinity is NOT a number and for the most part doesn’t behave like a number. However, despite that we’ll think of infinity in this section as a really, really, really large number that is so large there isn’t another number larger than it. This is not correct of course but may help with the discussion in this section. Note as well that everything that we’ll be discussing in this section applies only to real numbers. If you move into complex numbers for instance things can and do change.

So, let’s start thinking about addition with infinity. When you add two non-zero numbers you get a new number. For example, \(4 + 7 = 11\). With infinity this is not true. With infinity you have the following.

\[\begin{align*}\infty + a & = \infty \hspace{0.25in}{\mbox{where }}a \ne - \infty \\ \infty + \infty & = \infty \end{align*}\]In other words, a really, really large positive number (\(\infty \)) plus any positive number, regardless of the size, is still a really, really large positive number. Likewise, you can add a negative number (*i.e.* \(a < 0\)) to a really, really large positive number and stay really, really large and positive. So, addition involving infinity can be dealt with in an intuitive way if you’re careful. Note as well that the \(a\) must NOT be negative infinity. If it is, there are some serious issues that we need to deal with as we’ll see in a bit.

Subtraction with negative infinity can also be dealt with in an intuitive way in most cases as well. A really, really large negative number minus any positive number, regardless of its size, is still a really, really large negative number. Subtracting a negative number (*i.e.* \(a < 0\)) from a really, really large negative number will still be a really, really large negative number. Or,

Again, \(a\) must not be negative infinity to avoid some potentially serious difficulties.

Multiplication can be dealt with fairly intuitively as well. A really, really large number (positive, or negative) times any number, regardless of size, is still a really, really large number we’ll just need to be careful with signs. In the case of multiplication we have

\[\begin{array}{c}\left( a \right)\left( \infty \right) = \infty \hspace{0.25in}{\mbox{if }}a > 0\hspace{0.75in}\left( a \right)\left( \infty \right) = - \infty \hspace{0.25in}{\mbox{if }}a < 0\\ \\ \left( \infty \right)\left( \infty \right) = \infty \hspace{0.5in}\left( { - \infty } \right)\left( { - \infty } \right) = \infty \hspace{0.75in}\left( { - \infty } \right)\left( \infty \right) = - \infty \end{array}\]What you know about products of positive and negative numbers is still true here.

Some forms of division can be dealt with intuitively as well. A really, really large number divided by a number that isn’t too large is still a really, really large number.

\[\begin{align*}\frac{\infty }{a} & = \infty & \hspace{0.25in} & {\mbox{if }}a > 0,a \ne \infty & \hspace{0.75in}\frac{\infty }{a} & = - \infty & \hspace{0.25in}{\mbox{if }}a < 0,a \ne - \infty \\ \frac{{ - \infty }}{a} & = - \infty & \hspace{0.25in} & {\mbox{ if }}a > 0,a \ne \infty & \hspace{0.75in}\frac{{ - \infty }}{a} & = \infty & \hspace{0.25in}{\mbox{ if }}a < 0,a \ne - \infty \end{align*}\]Division of a number by infinity is somewhat intuitive, but there are a couple of subtleties that you need to be aware of. When we talk about division by infinity we are really talking about a limiting process in which the denominator is going towards infinity. So, a number that isn’t too large divided an increasingly large number is an increasingly small number. In other words, in the limit we have,

\[\frac{a}{\infty } = 0\hspace{0.5in}\hspace{0.5in}\hspace{0.25in}\frac{a}{{ - \infty }} = 0\]So, we’ve dealt with almost every basic algebraic operation involving infinity. There are two cases that that we haven’t dealt with yet. These are

\[\infty - \infty = {\mbox{?}}\hspace{0.5in}\hspace{0.5in}\frac{{ \pm \,\infty }}{{ \pm \,\infty }} = ?\]The problem with these two cases is that intuition doesn’t really help here. A really, really large number minus a really, really large number can be anything (\( - \infty \), a constant, or \(\infty \)). Likewise, a really, really large number divided by a really, really large number can also be anything (\( \pm \infty \) – this depends on sign issues, 0, or a non-zero constant).

What we’ve got to remember here is that there are really, really large numbers and then there are really, really, really large numbers. In other words, some infinities are larger than other infinities. With addition, multiplication and the first sets of division we worked this wasn’t an issue. The general size of the infinity just doesn’t affect the answer in those cases. However, with the subtraction and division cases listed above, it does matter as we will see.

Here is one way to think of this idea that some infinities are larger than others. This is a fairly dry and technical way to think of this and your calculus problems will probably never use this stuff, but it is a nice way of looking at this. Also, please note that I’m not trying to give a precise proof of anything here. I’m just trying to give you a little insight into the problems with infinity and how some infinities can be thought of as larger than others. For a much better (and definitely more precise) discussion see,

http://www.math.vanderbilt.edu/~schectex/courses/infinity.pdf

Let’s start by looking at how many integers there are. Clearly, I hope, there are an infinite number of them, but let’s try to get a better grasp on the “size” of this infinity. So, pick any two integers completely at random. Start at the smaller of the two and list, in increasing order, all the integers that come after that. Eventually we will reach the larger of the two integers that you picked.

Depending on the relative size of the two integers it might take a very, very long time to list all the integers between them and there isn’t really a purpose to doing it. But, it could be done if we wanted to and that’s the important part.

Because we could list all these integers between two randomly chosen integers we say that the integers are *countably infinite*. Again, there is no real reason to actually do this, it is simply something that can be done if we should choose to do so.

In general, a set of numbers is called countably infinite if we can find a way to list them all out. In a more precise mathematical setting this is generally done with a special kind of function called a bijection that associates each number in the set with exactly one of the positive integers. To see some more details of this see the pdf given above.

It can also be shown that the set of all fractions are also countably infinite, although this is a little harder to show and is not really the purpose of this discussion. To see a proof of this see the pdf given above. It has a very nice proof of this fact.

Let’s contrast this by trying to figure out how many numbers there are in the interval \( \left(0,1\right) \). By numbers, I mean all possible fractions that lie between zero and one as well as all possible decimals (that aren’t fractions) that lie between zero and one. The following is similar to the proof given in the pdf above but was nice enough and easy enough (I hope) that I wanted to include it here.

To start let’s assume that all the numbers in the interval \( \left(0,1\right) \) are countably infinite. This means that there should be a way to list all of them out. We could have something like the following,

\[\begin{align*}{x_1} & = 0.692096 \cdots \\ {x_2} & = 0.171034 \cdots \\ {x_3} & = 0.993671 \cdots \\ {x_4} & = 0.045908 \cdots \\ \vdots \,\, & \hspace{0.6in} \vdots \end{align*}\]Now, select the \(i\)^{th} decimal out of \({x_i}\) as shown below

and form a new number with these digits. So, for our example we would have the number

\[x = 0.6739 \cdots \]In this new decimal replace all the 3’s with a 1 and replace every other numbers with a 3. In the case of our example this would yield the new number

\[\overline x = 0.3313 \cdots \]Notice that this number is in the interval \( \left(0,1\right) \) and also notice that given how we choose the digits of the number this number will not be equal to the first number in our list, \({x_1}\), because the first digit of each is guaranteed to not be the same. Likewise, this new number will not get the same number as the second in our list, \({x_2}\), because the second digit of each is guaranteed to not be the same. Continuing in this manner we can see that this new number we constructed, \(\overline x \), is guaranteed to not be in our listing. But this contradicts the initial assumption that we could list out all the numbers in the interval \( \left(0,1\right) \). Hence, it must not be possible to list out all the numbers in the interval \( \left(0,1\right) \).

Sets of numbers, such as all the numbers in \( \left(0,1\right) \), that we can’t write down in a list are called *uncountably* infinite.

The reason for going over this is the following. An infinity that is uncountably infinite is significantly larger than an infinity that is only countably infinite. So, if we take the difference of two infinities we have a couple of possibilities.

\[\begin{align*}\infty \left( {{\mbox{uncountable}}} \right) - \infty \left( {{\mbox{countable}}} \right) & = \infty \\ & \\ \infty \left( {{\mbox{countable}}} \right) - \infty \left( {{\mbox{uncountable}}} \right) & = - \infty \end{align*}\]Notice that we didn’t put down a difference of two infinities of the same type. Depending upon the context there might still have some ambiguity about just what the answer would be in this case, but that is a whole different topic.

We could also do something similar for quotients of infinities.

\[\begin{align*}\frac{{\infty \left( {{\mbox{countable}}} \right)}}{{\infty \left( {{\mbox{uncountable}}} \right)}} & = 0\\ & \\ \frac{{\infty \left( {{\mbox{uncountable}}} \right)}}{{\infty \left( {{\mbox{countable}}} \right)}} & = \infty \end{align*}\]Again, we avoided a quotient of two infinities of the same type since, again depending upon the context, there might still be ambiguities about its value.

So, that’s it and hopefully you’ve learned something from this discussion. Infinity simply isn’t a number and because there are different kinds of infinity it generally doesn’t behave as a number does. Be careful when dealing with infinity.